As fullerenes, tubes, and cones are produced in the vapor phase we consider all three structures being originated by a similar-type nucleation seed, a small curved carbon sheet composed
Trang 168 K SATTLER
1u.4
20.8
Fig 3 STM images of fullerene tubes on a graphite
substrate
formed, further concentric shells can be added by gra-
phitic cylindrical layer growth
The c60+10i (j = 1,2,3, ) tube has an outer di-
ameter of 9.6 A[18] In its armchair configuration, the
hexagonal rings are arranged in a helical fashion with
a chiral angle of 30 degrees In this case, the tube axis
is the five-fold symmetric axis of the C60-cap The
single-shell tube can be treated as a rolled-up graph-
ite sheet that matches perfectly at the closure line
Choosing the cylinder joint in different directions
leads to different helicities One single helicity gives
a set of discrete diameters To obtain the diameter that
matches exactly the required interlayer spacing, the
tube layers need to adjust their helicities Therefore,
in general, different helicities for different layers in a
multilayer tube are expected In fact, this is confirmed
by our experiment
In Fig 4 we observe, in addition to the atomic lat-
tice, a zigzag superpattern along the axis at the sur-
face of the tube The zigzag angle is 120" and the
period is about 16 A Such superstructure (giant lat-
tice) was found earlier for plane graphite[lO] It is a
Moire electron state density lattice produced by dif-
ferent helicities of top shell and second shell of a tube
The measured period of 16 A reveals that the second
layer of the tube is rotated relative to the first layer
by 9" The first and the second cylindrical layers,
therefore, have chiral angles of 5" and -4", respec-
tively This proves that the tubes are, indeed, com-
posed of at least two coaxial graphitic cylinders with
different helicities
5 BUNDLES
In some regions of the samples the tubes are found
to be closely packed in bundles[20] Fig 5 shows a
-200 A broad bundle of tubes Its total length is
2000 A, as determined from a larger scale image The
diameters of the individual tubes range from 20-40 A
They are perfectly aligned and closely packed over the
whole length of the bundle
Our atomic-resolution studies[ 101 did not reveal
any steps or edges, which shows that the tubes are per-
*
4 0 0
3 0 0
2.00
1.00
0
0 1 0 0 2 0 0 3.00 4,00
nu
Fig 4 Atomic resolution STM image of a carbon tube, 35 A
in diameter In addition to the atomic structure, a zigzag su- perpattern along the tube axis can be seen
fect graphitic cylinders The bundle is disturbed in a small region in the upper left part of the image In the closer view in Fig 6, we recognize six tubes at the bun- dle surface The outer shell of each tube is broken, and
an inner tube is exposed We measure again an inter- tube spacing of -3.4 A This shows that the exposed inner tube is the adjacent concentric graphene shell The fact that all the outer shells of the tubes in the bundles are broken suggests that the tubes are strongly coupled through the outer shells The inner tubes, however, were not disturbed, which indicates that the
I
I
D
Fig 5 STM image of a long bundle of carbon nanotubes
The bundle is partially broken in a small area in the upper left part of the image Single tubes on the flat graphite sur-
face are also displayed
Trang 2STM of carbon nanotubes and nanocones 69
Fig 6 A closer view of the disturbed area of the bundle in
Fig 5 ; the concentric nature of the tubes is shown The outer-
most tubes are broken and the adjacent inner tubes are
complete
intertube interaction is weaker than the intratube in-
teraction This might be the reason for bundle forma-
tion in the vapor phase After a certain diameter is
reached for a single tube, growth of adjacent tubes
might be energetically favorable over the addition of
further concentric graphene shells, leading to the gen-
eration of bundles
6 CONES
Nanocones of carbon are found[3] in some areas
on the substrate together with tubes and other
mesoscopic structures In Fig 7 two carbon cones are
displayed For both cones we measure opening angles
of 19.0 f 0.5" The cones are 240 A and 130 A long
Strikingly, all the observed cones (as many as 10 in a
(800 A)2 area) have nearly identical cone angles - 19"
At the cone bases, flat or rounded terminations
were found The large cone in Fig 7 shows a sharp
edge at the base, which suggests that it is open The
small cone in this image appears closed by a spherical-
shaped cap
We can model a cone by rolling a sector of a sheet
around its apex and joining the two open sides If the
sheet is periodically textured, matching the structure
at the closure line is required to form a complete net-
work, leading to a set of discrete opening angles The
higher the symmetry of the network, the larger the set
In the case of a honeycomb structure, the sectors with
angles of n x (2p/6) ( n = 1,2,3,4,5) can satisfy per-
fect matching Each cone angle is determined by the
corresponding sector The possible cone angles are
19.2", 38.9", 60", 86.6", and 123.6", as illustrated in
Fig 8 Only the 19.2" angle was observed for all the
cones in our experiment A ball-and-stick model of the
19.2" fullerene cone is shown in Fig 9 The body part
Fig 7 A (244 A) STM image of two fullerene cones
is a hexagon network, while the apex contains five pentagons The 19.2" cone has mirror symmetry through a plane which bisects the 'armchair' and 'zig- zag' hexagon rows
It is interesting that both carbon tubules and cones have graphene networks A honeycomb lattice with- out inclusion of pentagons forms both structures However, their surface nets are configured differently The graphitic tubule is characterized by its diameter and its helicity, and the graphitic cone is entirely char- acterized by its cone angle Helicity is not defined for the graphitic cone The hexagon rows are rather ar- ranged in helical-like fashion locally Such 'local he- licity' varies monotonously along the axis direction of the cone, as the curvature gradually changes One can
Fig 8 The five possible graphitic cones, with cone angles
of 19.2", 38.9", 60", 86.6", and 123.6"
Trang 370 K SATTLER
a r m c h a i r
c
apex
z i g z a g
Fig 9 Ball-and-stick model for a 19.2" fullerene cone The
back part of the cone is identical to the front part displayed
in the figure, due to the mirror symmetry The network is in
'armchair' and 'zigzag' configurations, at the upper and lower
sides, respectively The apex of the cone is a fullerene-type
cap containing five pentagons
easily show that moving a 'pitch' (the distance between
two equivalent sites in the network) along any closure
line of the network leads to another identical cone For
the 19.2" cone, a hexagon row changes its 'local heli-
cal' direction at half a turn around the cone axis and
comes back after a full turn, due to its mirror symme-
try in respect to its axis The other four cones, with
larger opening angles, have Dnd (n = 2,3,4,5) symme-
try along the axis The 'local helicity' changes its di-
rection at each of their symmetry planes
As fullerenes, tubes, and cones are produced in the
vapor phase we consider all three structures being
originated by a similar-type nucleation seed, a small
curved carbon sheet composed of hexagons and pent-
agons The number of pentagons (m) in this fullerene-
type (m-P) seed determines its shape Continuing growth
of an alternating pentagodhexagon (516) network
leads to the formation of C60 (and higher fullerenes)
If however, after the Cb0 hemisphere is completed,
growth continues rather as a graphitic ( 6 / 6 ) network,
a tubule is formed If graphitic growth progresses
from seeds containing one to five pentagons, fuller-
ene cones can be formed
The shape of the 5-P seed is closest to spherical
among the five possible seeds Also, its opening an-
gle matches well with the 19.2" graphitic cone There-
fore, continuing growth of a graphitic network can
proceed from the 5-P seed, without considerable strain
in the transition region The 2-P, 3-P, and 4-P seeds
would induce higher strain (due to their nonspherical
shapes) to match their corresponding cones and are unlikely to form This explains why only the 19.2"
cones have been observed in our experiment Carbon cones are peculiar mesoscopic objects They are characterized by a continuous transition from fullerene to graphite through a tubular-like in- termedium The dimensionality changes gradually as
the cone opens It resembles a 0-D cluster at the apex,
then proceeds to a 1-D 'pipe' and finally approaches
a 2-D layer The cabon cones may have complex band
structures and fascinating charge transport properties, from insulating at the apex to metallic at the base They might be used as building units in future nano- scale electronics devices
Acknowledgements-Financial support from the National Science Foundation, Grant No DMR-9106374, is gratefully
acknowledged
REFERENCES
1 S Iijima, Nature 354, 56 (1991) T W Ebbesen and P
2 D Ugarte, Nature 359, 707 (1992)
3 M Ge and K Sattler, Chem Phys Lett 220, 192 (1994)
4 M Ge and K Sattler, Science 260, 515 (1993)
5 T W Ebbesen, H Hiura, J Fijita, Y Ochiai, S Mat-
sui, and K Tanigaki, Chem Phys Lett 209,83 (1993)
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Trang 4TOPOLOGICAL AND SP3 DEFECT STRUCTURES
IN NANOTUBES
T W EBBESEN' and T TAKADA~
'NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, U.S.A
2Fundamental Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba 305, Japan
(Received 25 November 1994; accepted 10 February 1995)
Abstract-Evidence is accumulating that carbon nanotubes are rarely as perfect as they were once thought
to be Possible defect structures can be classified into three groups: topological, rehybridization, and in- complete bonding defects The presence and significance of these defects in carbon nanotubes are discussed
It is clear that some nanotube properties, such as their conductivity and band gap, will be strongly affected
by such defects and that the interpretation of experimental data must be done with great caution
Key Words-Defects, topology, nanotubes, rehybridization
1 INTRODUCTION
Carbon nanotubes were first thought of as perfect
seamless cylindrical graphene sheets - a defect-free
structure However, with time and as more studies
have been undertaken, it is clear that nanotubes are
not necessarily that perfect; this issue is not simple be-
cause of a variety of seemingly contradictory observa-
tions The issue is further complicated by the fact that
the quality of a nanotube sample depends very much
on the type of machine used to prepare it[l] Although
nanotubes have been available in large quantities since
1992[2], it is only recently that a purification method
was found[3] So, it is now possible to undertakevarious
accurate property measurements of nanotubes How-
ever, for those measurements to be meaningful, the
presence and role of defects must be clearly understood
The question which then arises is: What do we call
a defect in a nanotube? To answer this question, we
need to define what would be a perfect nanotube
Nanotubes are microcrystals whose properties are
mainly defined by the hexagonal network that forms
the central cylindrical part of the tube After all, with
an aspect rat.io (length over diameter) of 100 to 1000,
the tip structure will be a small perturbation except
near the ends This is clear from Raman studies[4] and
is also the basis for calculations on nanotube proper-
ties[5-71 So, a perfect nanotube would be a cylindrical
graphene sheet composed only of hexagons having a
minimum of defects at the tips to form a closed seam-
less structure
Needless to say, the issue of defects in nanotubes
is strongly related to the issue of defects in graphene
Akhough earlier studies of graphite help us under-
stand nanotubes, the concepts derived from fullerenes
has given us a new insight into traditional carbon ma-
terials So, the discussion that follows, although aimed
at nanotubes, is relevant to all graphitic materials
First, different types of possible defects are described
Then, recent evidences for defects in nanotubes and
their implications are discussed
2 CLASSES OF DEFECTS
Figure 1 show examples of nanotubes that are far from perfect upon close inspection They reveal some
of the types of defects that can occur, and will be dis- cussed below Having defined a perfect nanotube as
a cylindrical sheet of graphene with the minimum number of pentagons at each tip to form a seamless structure, we can classify the defects into three groups: 1) topological defects, 2) rehybridization defects and 3) incomplete bonding and other defects Some defects will belong to more than one of these groups, as will
be indicated
2.1 Topological defects
The introduction of ring sizes other than hexagons, such as pentagons and heptagons, in the graphene sheet creates topological changes that can be treated
as local defects Examples of the effect of pentagons and heptagons on the nanotube structure is shown in Fig 1 (a) The resulting three dimensional topology follows Euler's theorem[8] in the approximation that
we assume that all the individual rings in the sheet are flat In other words, it is assumed that all the atoms
of a given cycle form a plane, although there might
be angles between the planes formed in each cycle In reality, the strain induced by the three-dimensional ge- ometry on the graphitic sheet can lead to deformation
of the rings, complicating the ideal picture, as we shall see below
From Euler's theorem, one can derive the follow- ing simple relation between the number and type of cycles nj (where the subscript i stands for the number
of sides to the ring) necessary to close the hexagonal network of a graphene sheet:
3n, + 2n, + n5 - n7 - 2n8 - 372, = 12 where 12 corresponds to a total disclination of 4n (i.e.,
a sphere) For example, in the absence of other cycles one needs 12 pentagons (ns) in the hexagonal net-
71
Trang 512 T W EBBESEN and T TAKADA
Fig 1 Five examples of nanotubes showing evidence of defects in their structure (p: pentagon, h: hep-
tagon, d: dislocation); see text (the scale bars equal 10 nm)
work to close the structure The addition of one hep-
tagon (n7) to the nanotube will require the presence
of 13 pentagons to close the structure (and so forth)
because they induce opposite 60" disclinations in the
surface Although the presence of pentagons (ns) and
heptagons (n,) in nanotubes[9,10] is clear from the
disclinations observed in their structures (Fig la), we
are not aware of any evidence for larger or smaller cy-
cles (probably because the strain would be too great)
A single heptagon or pentagon can be thought of
as point defects and their properties have been calcu-
lated[l I] Typical nanotubes don't have large numbers
of these defects, except close to the tips However, the
point defects polygonize the tip of the nanotubes, as
shown in Fig 2 This might also favor the polygonal-
ization of the entire length of the nanotube as illus-
trated by the dotted lines in Fig 2 Liu and Cowley
have shown that a large fraction of nanotubes are po-
lygonized in the core[ 12,131 This will undoubtedly have
significant effects on their properties due to local re-
hybridization, as will be discussed in the next section
The nanotube in Fig 1 (e) appears to be polygonized
(notice the different spacing between the layers on the left and right-hand side of the nanotube)
Another common defect appears to be the aniline structure that is formed by attaching a pentagon and
a heptagon to each other Their presence is hard to de- tect directly because they create only a small local de- formation in the width of the nanotube However, from time to time, when a very large number of them are accidentally aligned, the nanotube becomes grad- ually thicker and thicker, as shown in Fig 1 (b) The existence of such tubes indicates that such pairs are probably much more common in nanotubes, but that they normally go undetected because they cancel each other out (random alignment) The frequency of oc- currence of these aligned 5/7 pairs can be estimated
to be about 1 per 3 nm from the change in the diame-
ter of the tube Randomly aligned 5 / 7 pairs should be
present at even higher frequencies, seriously affecting the nanotube properties Various aspects of such pairs have been discussed from a theoretical point of view
in the literature[l4,15] In particular, it has been pointed out by Saito et a1.[14] that such defect pairs
Trang 6Topological and sp3 defect structures in nanotubes 13
Fig 1 continued
can annihilate, which would be relevant to the anneal-
ing away of such defects at high temperature as dis-
cussed in the next section
It is not possible to exclude the presence of other
unusual ring defects, such as those observed in graphitic
sheets[ 16,171 For example, there might be heptagon-
triangle pairs in which there is one sp3 carbon atom
bonded to 4 neighboring atoms, as shown in Fig 3
Although there must be strong local structural distor-
tions, the graphene sheet remains flat overa11[16,17]
This is a case where Euler’s theorem does not apply
The possibility of sp3 carbons in the graphene sheet
brings us to the subject of rehybridization
2.2 Rehybridization defects
The root of the versatility of carbon is its ability
to rehybridize between sp, s p 2 , and s p 3 While dia-
mond and graphite are examples of pure sp3 and sp2
hybridized states of carbon, it must not be forgotten
that many intermediate degrees of hybridization are
possible This allows for the out-of-plane flexibility of
graphene, in contrast to its extreme in-plane rigidity
As the graphene sheet is bent out-of-plane, it must lose
some of its sp2 character and gain some sp3 charac-
ter or, to put it more accurately, it will have s p Z f L V
character The size of CY will depend on the degree of curvature of the bend The complete folding of the graphene sheet will result in the formation of a defect line having strong sp3 character in the fold
We have shown elsewhere that line defects having
sp3 character form preferentially along the symmetry
axes of the graphite sheet[lb] This is best understood
by remembering that the change from sp2 to sp3 must naturally involve a pair of carbon atoms because a double bond is perturbed In the hexagonal network
of graphite shown in Fig 4, it can be seen that there are 4 different pairs of carbon atoms along which the
sp3 type line defect can form Two pairs each are
found along the [loo] and [210] symmetry axes Fur- thermore, there are 2 possible conformations, “boat” and “chair,” for three of these distinct line defects and
a single conformation of one of them These are illus- trated in Fig 5
In the polygonized nanotubes observed by Liu and Cowley[12,13], the edges of the polygon must have more sp3 character than the flat faces in between
These are defect lines in the sp2 network Nanotubes
mechanically deformed appear to be rippled, indicat-
Trang 774 T W EBBESEN and T TAKADA
I
Fig 1 continued
ing the presence of ridges with sp3 character[l8] Be-
cause the symmetry axes of graphene and the long axis
of the nanotubes are not always aligned, any defect
line will be discontinuous on the atomic scale as it tra-
verses the entire length of the tube Furthermore, in
the multi-layered nanotubes, where each shell has a
different helicity, the discontinuity will not be super-
imposable In other words, in view of the turbostratic
nature of the multi-shelled nanotubes, an edge along
the tube will result in slightly different defect lines in
each shell
/
/
/ /
/
/
/
,
/
/
/
/ / /
Fig 2 Nanotube tip structure seen from the top; the pres- ence of pentagons can clearly polygonize the tip
2.3 Incomplete bonding and other defects
Defects traditionally associated with graphite might also be present in nanotubes, although there is not yet much evidence for their presence For instance, point defects such as vacancies in the graphene sheet might
be present in the nanotubes Dislocations are occasion- ally observed, as can be seen in Fig 1 (c) and (d), but they appear to be quite rare for the nanotubes formed
at the high temperatures of the carbon arc It might
be quite different for catalytically grown nanotubes
In general, edges of graphitic domains and vacancies should be chemically very reactive as will be discussed below
3 DISCUSSION
There are now clear experimental indications that nanotubes are not perfect in the sense defined in the
introduction[l2,13,19,20] The first full paper dedi- cated to this issue was by Zhou et al.[19], where both pressure and intercalation experiments indicated that the particles in the sample (including nanotubes) could not be perfectly closed graphitic structures It was pro-
Fig 3 Schematic diagram of heptagon-triangle defects
[ 16,171
Trang 8Topological and sp3 defect structures in nanotubes 15
Fig 4 Hexagonal network of graphite and the 4 different
pairs of carbon atoms across which the sp3-like defect line
may form[l8]
posed that nanotubes were composed of pieces of gra-
phitic sheets stuck together in a paper-machi model
The problem with this model is that it is not consis-
tent with two other observations First, when nano-
tubes are oxidized they are consumed from the tip
Fig 5 Conformations of the 4 types of defect lines that can
occur in the graphene sheet[l8]
inwards, layer by layer[21,22] If there were smaller domains along the cylindrical part, their edges would
be expected to react very fast to oxidation, contrary
to observation Second, ESR studies[23] do not reveal any strong signal from dangling bonds and other de- fects, which would be expected from the numerous edges in the paper-machk model
To try to clarify this issue, we recently analyzed crude nanotube samples and purified nanotubes be- fore and after annealing them at high temperature[20]
It is well known that defects can be annealed away at high temperatures (ca 285OOC) The annealing effect was very significant on the ESR properties, indicating clearly the presence of defects in the nanotubes[20] However, our nanotubes do not fit the defect struc- ture proposed in the paper-machi model for the rea- son discussed in the previous paragraph Considering the types of possible defects (see part 2), the presence
of either a large number of pentagon/heptagon pairs
in the nanotubes and/or polygonal nanotubes, as ob- served by Liu and Cowley[12,13], could possibly ac- count for these results Both the 5/7 pairs and the edges of the polygon would significantly perturb the electronic properties of the nanotubes and could be annealed away at very high temperatures The sensi- tivity of these defects to oxidation is unknown
In attempting to reconcile these results with those
of other studies, one is limited by the variation in sam- ple quality from one study to another For instance, IESR measurements undertaken on bulk samples in three different laboratories shoq7 very different re- sults[19,23,24] As we have pointed out elsewhere, the
quantity of nanotubes (and their quality) varies from
a few percent to over 60% of the crude samples, de- pending on the current control and the extent of cool- ing in the carbon arc apparatus[l] The type and distribution of defects might also be strongly affected
by the conditions during nanotube production The ef- fect of pressure on the spacing between the graphene sheets observed by Zhou et al argues most strongly
in favor of the particles in the sample having a non-
closed structure[l9] Harris et af actually observe that nanoparticles in these samples sometimes d o not form closed structures[25] It would be interesting to repeat the pressure study on purified nanotubes before and after annealing with samples of various origins This should give significant information on the nature of the defects The results taken before annealing will,
no doubt, vary depending on where and how the sam- ple was prepared The results after sufficient anneal- ing should be consistent and independent of sample origin
4 CONCLUSION
The issue of defects in nanotubes is very important
in interpreting the observed properties of nanotubes For instance, electronic and magnetic properties will
be significantly altered as is already clear from obser- vation of the conduction electron spin resonance[20,23]
Trang 976
It would be worthwhile making theoretical calculations
T W EBBESEN and T TAKADA
11 R Tamura and M Tsukada Phvs Rev B 49 7697
-
to evaluate the effect of defects on the nanotube prop-
erties The chemistry might be affected, although to
a lesser degree because nanotubes, like graphite, are
chemically quite inert If at all possible, nanotubes
should be annealed (if not also purified) before phys-
ical measurements are made Only then are the results
likely to be consistent and unambiguous
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Trang 10HELICALLY COILED AND TOROIDAL CAGE FORMS
OF GRAPHITIC CARBON
SIGEO IHARA and SATOSHI ITOH
Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo 185, Japan
(Received 22 August 1994; accepted in revised form 10 February 1995)
Abstract-Toroidal forms for graphitic carbon are classified into five possible prototypes by the ratios
of their inner and outer diameters, and the height of the torus Present status of research of helical and toroidal forms, which contain pentagons, hexagons, and heptagons of carbon atoms, are reviewed By molecular-dynamics simulations, we studied the length and width dependence of the stability of the elon-
gated toroidal structures derived from torus C240 and discuss their relation to nanotubes The atomic ar-
rangements of the structures of the helically coiled forms of the carbon cage for the single layer, which are found to be thermodynamically stable, are compared to those of the experimental helically coiled forms
of single- and multi-layered graphitic forms that have recently been experimentally observed
Key Words-Carbon, molecular dynamics, torus, helix, graphitic forms
1 INTRODUCTION
Due, in part, to the geometrical uniqueness of their
cage structure and, in part, to their potentially tech-
nological use in various fields, fullerenes have been the
focus of very intense research[l] Recently, higher
numbers of fullerenes with spherical forms have been
available[2] It is generally recognized that in the ful-
lerene, C60, which consists of pentagons and hexa-
gons formed by carbon atoms, pentagons play an
essential role in creating the convex plane This fact
was used in the architecture of the geodesic dome in-
vented by Robert Buckminster Fuller[3], and in tra-
ditional bamboo art[4] (‘toke-zaiku’,# for example)
By wrapping a cylinder with a sheet of graphite, we
can obtain a carbon nanotube, as experimentally ob-
served by Iijima[S] Tight binding calculations indicate
that if the wrapping is charged (i.e., the chirality of the
surface changes), the electrical conductivity changes:
the material can behave as a semiconductor or metal
depending on tube diameter and chirality[6]
In the study of the growth of the tubes, Iijima
found that heptagons, seven-fold rings of carbon at-
oms, appear in the negatively curved surface Theo-
retically, it is possible to construct a crystal with only
a negatively curved surface, which is called a minimal
surface[7] However, such surfaces of carbon atoms
are yet to be synthesized The positively curved sur-
face is created by insertion of pentagons into a hex-
agonal sheet, and a negatively curved surface is created
by heptagons Combining these surfaces, one could,
in principle, put forward a new form of carbon, hav-
ing new features of considerable technological inter-
est by solving the problem of tiling the surface with
pentagons, heptagons, and hexagons
#At the Ooishi shrine of Ako in Japan, a geodesic dome
made of bamboo with three golden balls, which was the sym-
bol called “Umajirushi” used by a general named Mori Mis-
aemon’nojyo Yoshinari at the battle of Okehazama in 1560,
has been kept in custody (See ref 141)
The toroidal and helical forms that we consider here are created as such examples; these forms have quite interesting geometrical properties that may lead
to interesting electrical and magnetic properties, as well as nonlinear optical properties Although the method of the simulations through which we evaluate the reality of the structure we have imagined is omit- ted, the construction of toroidal forms and their prop- erties, especially their thermodynamic stability, are discussed in detail Recent experimental results on to- roidal and helically coiled forms are compared with theoretical predictions
2 TOPOLOGY OF TOROIDAL AND HELICAL FORMS
2.1 Tiling rule f o r cage structure
of graphitic carbon
Because of the sp2 bonding nature of carbon at-
oms, the atoms on a graphite sheet should be con- nected by the three bonds Therefore, we consider how
to tile the hexagons created by carbon atoms on the toroidal surfaces Of the various bonding lengths that can be taken by carbon atoms, we can tile the toroi- dal surface using only hexagons Such examples are provided by Heilbonner[8] and Miyazakif91 However, the side lengths of the hexagons vary substantially If
we restrict the side length to be almost constant as in graphite, we must introduce, at least, pentagons and heptagons
Assuming that the surface consists of pentagons, hexagons, and heptagons, we apply Euler’s theorem Because the number of hexagons is eliminated by a kind of cancellation, the relation thus obtained con- tains only the number of pentagons and heptagons:
fs - f, = 12(1-g), where fs stands for the number of
pentagons, f, the number of heptagons, and g is the genius (the number of topological holes) of the surface
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